A050322 Number of factorizations indexed by prime signatures: A001055(A025487).

1, 1, 2, 2, 3, 4, 5, 7, 5, 7, 9, 12, 11, 11, 16, 19, 21, 15, 29, 26, 30, 15, 31, 38, 22, 47, 52, 45, 36, 57, 64, 30, 77, 98, 67, 74, 97, 66, 105, 42, 109, 118, 92, 109, 171, 97, 141, 162, 137, 165, 56, 212, 181, 52, 198, 189, 289, 139, 250, 257, 269, 254, 77, 382, 267

Offset

1,3

Comments

For A025487(m) = 2^k = A000079(k), we have a(m) = A000041(k).

Is a(k) = A000110(k) for A025487(m) = A002110(k)?

Links

R. J. Mathar and Michael De Vlieger, Table of n, a(n) for n = 1..5000 (First 300 terms from R. J. Mathar)

R. E. Canfield, P. Erdős and C. Pomerance, On a Problem of Oppenheim concerning "Factorisatio Numerorum", J. Number Theory 17 (1983), 1-28.

Jun Kyo Kim, On highly factorable numbers, Journal Of Number Theory, Vol. 72, No. 1 (1998), pp. 76-91.

Example

From Gus Wiseman, Jan 13 2020: (Start)
The a(1) = 1 through a(11) = 9 factorizations:
  {}  2  4    6    8      12     16       24       30     32         36
         2*2  2*3  2*4    2*6    2*8      3*8      5*6    4*8        4*9
                   2*2*2  3*4    4*4      4*6      2*15   2*16       6*6
                          2*2*3  2*2*4    2*12     3*10   2*2*8      2*18
                                 2*2*2*2  2*2*6    2*3*5  2*4*4      3*12
                                          2*3*4           2*2*2*4    2*2*9
                                          2*2*2*3         2*2*2*2*2  2*3*6
                                                                     3*3*4
                                                                     2*2*3*3
(End)

Maple

A050322 := proc(n)
    A001055(A025487(n)) ;
end proc: # R. J. Mathar, May 25 2017

Mathematica

c[1, r_] := c[1, r] = 1; c[n_, r_] := c[n, r] = Module[{d, i}, d = Select[Divisors[n], 1 < # <= r &]; Sum[c[n/d[[i]], d[[i]]], {i, 1, Length[d]}]]; Map[c[#, #] &, Union@ Table[Times @@ MapIndexed[If[n == 1, 1, Prime[First@ #2]]^#1 &, Sort[FactorInteger[n][[All, -1]], Greater]], {n, Product[Prime@ i, {i, 6}]}]] (* Michael De Vlieger, Jul 10 2017, after Dean Hickerson at A001055 *)
facs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]];
Length/@facs/@First/@GatherBy[Range[1000], If[#==1, {}, Sort[Last/@FactorInteger[#]]]&] (* Gus Wiseman, Jan 13 2020 *)

Crossrefs

Cf. A000041, A000079, A000110, A001055, A002110, A025487.

The version indexed by unsorted prime signature is A331049.

The version indexed by prime shadow (A181819, A181821) is A318284.

This sequence has range A045782 (same as A001055).

Cf. A033833, A045778, A045783, A070175, A181821, A325238, A330972, A330973, A330976, A330989, A330990, A330998, A331050.

Sequence in context: A045475 A316078 A082543 * A325512 A211858 A349149

Adjacent sequences: A050319 A050320 A050321 * A050323 A050324 A050325

Keyword

nonn

Author

Christian G. Bower, Oct 15 1999

Status

approved